Using sums of squares to prove that certain

entirefunctionshaveonlyrealzeros

by

GeorgeGasper1

DedicatedtothememoryofRalphP.Boas,Jr.(1912–1992)(PublishedinFourierAnalysis:AnalyticandGeometricAspects,W.O.Bray,P.S.MilojevicandC.V.Stanojevic,eds.,MarcelDekker,1994,pp.

171–186)

Abstract

ItisshownhowsumsofsquaresofrealvaluedfunctionscanbeusedtogivenewproofsoftherealityofthezerosoftheBesselfunc-tionsJα(z)whenα≥−1,confluenthypergeometricfunctions0F1(c;z)whenc>0or0>c>−1,LaguerrepolynomialsLαn(z)whenα≥−2,

(α,β)

andJacobipolynomialsPn(z)whenα≥−1andβ≥−1.Besidesyieldingnewinequalitiesfor|F(z)|2,whereF(z)isoneofthesefunc-tions,thederivedidentitiesleadtoinequalitiesfor∂|F(z)|2/∂yand∂2|F(z)|2/∂y2,whichalsogivenewproofsoftherealityofthezeros.

1Introduction

Ina1975surveypaper[11]onpositivityandspecialfunctionsitwasshownhowsumsofsquaresofspecialfunctionscouldbeusedtoprovethenonneg-ativityoftheFej´erkernel,thepositivityofintegralsofBesselfunctions[10]

SupportedinpartbytheNationalScienceFoundationundergrantDMS-9103177.Keywords.Entirefunctions,inequalities,realzeros,sumsofsquares,confluenthyper-geometricfunctions,Besselfunctions,Jacobipolynomials,Laguerrepolynomials.

1

1

andoftheCotes’numbersforsomeJacobiabscissas,aTur´antypeinequalityforBesselfunctions,theAskey-Gasperinequality(cf.[1],[2],[13],[15])(1.1)

n󰀃k=0

Pk

(α,0)

(x)≥0,α>−2,−1≤x≤1,

whichdeBranges[5]employedtocompletehisproofoftheBieberbachcon-jecture,andtoprovethemoregeneralinequalities[12](1.2)

(α,β)n󰀃(λ+1)k(λ+1)n−kPk(x)k=0

(β,α)(n−k)!Pk(1)

k!

≥0,−1≤x≤1,

when0≤λ≤α+βandβ≥−1/2.Itwasalsopointedoutin[11]that,

sinceoneofJensen’snecessaryandsufficientconditionsfortheRiemannHypothesistohold(giveninP´olya[19])istheconditionthat

󰀄

(1.3)

∞−∞

󰀄

∞−∞

Φ(s)Φ(t)ei(s+t)x(s−t)2ndsdt≥0,

−∞forn=0,1,2,3,...,where(1.4)

Φ(t)=2

∞󰀃k=1

(2k4π2e9t/2−3k2πe5t/2)e−k

2πe2t

,

andtheaboveintegralisasquarewhenn=0,themethodofsumsofsquaresissuggestedforproving(1.3).

AnotherofJensen’snecessaryandsufficientconditionsfortheRiemannHypothesistoholdisthat(1.5)󰀄

∞−∞

󰀄

∞−∞

Φ(s)Φ(t)ei(s+t)xe(s−t)y(s−t)2dsdt≥0,

−∞whichcanalsobewrittenintheequivalentform(1.6)with(1.7)

Ξ(z)=

∂2

|Ξ(x+iy)|2≥0,2∂y

󰀄

∞−∞

−∞󰀄

Φ(t)exp(izt)dt=2

2

∞0

Φ(t)cos(zt)dt.

That(1.6)isasufficientconditionfortheRiemannΞ(z)functiontohaveonlyrealzerosfollowsdirectlyfromobservationthat,since|Ξ(x+iy)|2=Ξ(x+iy)Ξ(x−iy)isanonnegativeevenfunctionofy,(1.6)impliesthat|Ξ(x+iy)|2isanonnegativeevenconvexfunctionofywithitsuniqueminimumaty=0,andhenceΞ(x+iy)=0whenevery=0.IfthefunctionΦ(t)in(1.3)and(1.5)isreplacedbyafunctionΨ(t)suchthattheconditionsstatedin[19,§1],aresatisfied,then,by[19,pp.17,18],theinequalitiesin(1.3)and(1.5)arenecessaryandsufficientconditionsfortheFourier(orcosine)transformofΨ(t)tohaveonlyrealzeros.In1913Jensen[18]provedthateachoftheinequalities(1.8)

∂

y|F(x+iy)|2≥0,∂y

∂22

|F(x+iy)|≥0,∂y2−∞isnecessaryandsufficientforarealentirefunctionF(z)≡0ofgenus0or1(cf.Boas[4,Chapter2])tohaveonlyrealzeros.AlsoseeTitchmarsh[21]andVarga[22,Chapter3].

Inviewoftheseobservationsandthesuccessesofthesumsofsquaresmethod(alsosee[14],[16,Chapter8]),sincetheearly1970’sIhavebeeninvestigatinghowsquaresofrealvaluedfunctionscanbeusedtoprovethatcertainentirefunctionshaveonlyrealzerosandtoproveinequalitiesoftheformin(1.8).InthispaperIdemonstratehowcertainseriesexpansionsinsumsofsquaresofspecialfunctionsgivenewproofsoftherealityofthezerosoftheBesselfunctionsJα(z)whenα≥−1,confluenthypergeometricfunctions0F1(c;z)whenc>0or0>c>−1,LaguerrepolynomialsLαn(z)

(α,β)

whenα≥−2,andJacobipolynomialsPn(z)whenα≥−1andβ≥−1.Here,aselsewhere,z=x+iyisacomplexvariableandxandyarerealvariables.Forthedefinitionsofthesefunctionsandtheirproperties,seeErd´elyi[9]andSzeg˝o[20].Inaddition,itwillbeshownthatbesidesyieldingnewinequalitiesfor|F(z)|2,whereF(z)isoneofthesefunctions,thederivedidentitiesleadtoinequalitiesfor∂|F(z)|2/∂yand∂2|F(z)|2/∂y2,whichalsogivenewproofsoftherealityofthezeros.

3

2Initialobservations

Inordertoseehoweasilysumsofsquarescanbeusedtoprovethatallofthezerosofsinzandcoszarereal,itsufficestoobservethatwehavethe(easilyverified)identities(2.1)(2.2)

|sinz|2=sin2x+sinh2y,|cosz|2=cos2x+sinh2y

andtonotethatsinhy=(ey−e−y)/2>0wheny>0,andsinhy<0wheny<0.

Onecanalsotakepartialderivativesoftheidentitiesin(2.1)and(2.2)withrespecttoytoobtain(2.3)

∂∂|sinz|2=|cosz|2=sinh2y∂y∂y

whichshowsthat|sinz|2and|cosz|2areincreasing(decreasing)functions

ofywheny>0(y<0),andtoobtain

∂2∂22

(2.4)|sinz|=2|cosz|2=2cosh2y=2(cosh2y+sinh2y)≥2,2∂y∂ywhichshowsthat|sinz|2and|cosz|2areconvexfunctionsofy.Then,be-cause|sinz|2and|cosz|2arenonnegativeevenfunctionsofy,itimmediatelyfollowsfrom(2.3)and(2.4)thatsinzandcoszhaveonlyrealzeros.

Observethattherealityofthezerosofsinzandcoszalsofollowsfromtheinequalities(2.5)(2.6)

|sinz|2>sin2x,

|cosz|2>cos2x,

(y=0)

|sinz|2≥sinh2y,|cosz|2≥sinh2y,

(2.7)

∂∂2

y|sinz|=y|cosz|2≥2y2,∂y∂y

4

(2.8)

∂2∂2222

|sinz|=|cosz|≥2coshy∂y2∂y2∂2∂22

|sinz|=2|cosz|2≥2+2sinh2y2∂y∂y

(2.9)

whichareconsequencesof(2.1)–(2.4).

3

Besselfunctionsand0F1(c;z)functions

Sincetheidentitiesandinequalitiesin§2givetherealityofthezerosofthe

Besselfunctions[20,(1.71.2)](3.1)

J−1(z)=(

221)2cosz,πz

J1(z)=(

221)2sinz,πz

thissuggeststhatitshouldbepossibletousesumsofsquarestoproveLommel’stheorem(seeWatson[23,p.482])thatallofthezerosoftheBesselfunction[9,7.2(3)](3.2)

(z/2)α2

Jα(z)=0F1(α+1;−z/4)

Γ(α+1)

arerealwhenα>−1.Withthisaiminmindandinordertoworkwithentirefunctions,weset(3.3)

Jα(z)=z

−α

2−α2

Jα(z)=0F1(α+1;−z/4),

Γ(α+1)

whichisanevenentirefunctionofzsuchthatJα(z)=Jα(z)whenαisreal.Letα>−1.Then,fromtheproductformula(37)inCarlitz[8],(3.4)

(α+1)2k−α

2k

|Jα(z)|=(x2+y2)kJα+k(2x).

k=0k!(2α+1)kΓ(α+1)

2

∞󰀃

ToexpresseachoftheBesselfunctionsontherightsideof(3.4)asasum

ofsquaresofrealvaluedBesselfunctionsobservethatfromtheaddition

5

theoremforBesselfunctions[9,7.15(30)]wehavetheexpansion(3.5)

Jα+k(2x)=2

k+α

∞󰀃(j+k+α)(2k+2α)jj=0

Γ(k+α)

j!

(−1)jx2j(Jα+j+k(x))2.

Hence,substituting(3.5)into(3.4)andchangingtheorderofsummationwe

findthat(3.6)|Jα(z)|=

2

∞󰀃(n+α)(2α)nn=0

αn!

(−1)nx2n

×2F1(−n,n+2α;2α+1;1+y2/x2)(Jα+n(x))2.

NowapplytheEulertransformationformula[9,2.9(3)](3.7)

2F1(a,b;c;z)

=(1−z)−a2F1(a,c−b;c;z/(z−1))

totheabove2F1seriestoobtainthedesiredsumofsquaresexpansionformula(3.8)

|Jα(z)|2=(Jα(x))2+2(α+1)y2(Jα+1(x))2

+

∞󰀃(2n+2α)(2α+1)n−1n=2

n!

y2n

×2F1(−n,1−n;2α+1;1+x2/y2)(Jα+n(x))2.

Whenn≥2,α>−1andy=0,thepositivityofthecoefficientsof

(Jα+n(x))2in(3.8)followsfrom

(3.9)(2α+1)2F1(−n,1−n;2α+1;1+x2/y2)=(2α+1+n2−n)

+n(n−1)x/y+

2

2

n󰀃(−n)k(1−n)kk=2

k!(2α+2)k−1

(1+x2/y2)k>0.

Hence,sincetherealzerosofJα(x)andJα+1(x)areinterlaced,(3.8)gives

asumofsquaresproofthattheentirefunctionsJα(z),andthustheBesselfunctionsJα(z),haveonlyrealzeroswhenα>−1.Lettingα→−1itfollowsthattheBesselfunctionJ−1(z)=limα→−1Jα(z)=−J1(z)hasonlyrealzeros.

Noticethattheinequality(3.10)|Jα(z)|2≥(Jα(x))2+2(α+1)(yJα+1(x))2>0,y=0,α>−1,

6

andinfactinfinitelymanyinequalitiesfollowfrom(3.8)byjustdroppingtermsfromtherightsideof(3.8).Analogousto(2.7)–(2.9),itfollowsbydifferentiatingequation(3.6)withrespecttoyandapplying(3.7)thatwealsohavetheidentities

∞󰀃∂(n+α+1)(2α+2)n2n22

(3.11)y|Jα(z)|=4yy

∂yn!n=0

×2F1(−n,−n;2α+2;1+x2/y2)(Jα+n+1(x))2

and(3.12)

∞󰀃(n+α+1)(2α+2)n2n∂22

y|J(z)|=4α

∂y2n!n=0

×

2F1(−n,−n;2α2

+2;1+x2/y2)(Jα+n+1(x))2n!

y2n

+8y×

∞󰀃(n+α+2)(2α+3)n+1n=0

2F1(−n,−n

−1;2α+3;1+x2/y2)(Jα+n+2(x))2,

whichgiveinfinitelymanyinequalities,suchas,e.g.,(3.13)(3.14)

y

∂

|Jα(z)|2≥4(α+1)(yJα+1(x))2≥0,∂y

α≥−1,α≥−1,

∂222

|J(z)|≥4(α+1)(J(x))≥0,αα+1∂y2eachofwhichprovesthatJα(z)hasonlyrealzeroswhenα≥−1.

Inviewof(3.3)therealityofthezerosofJα(z)whenα>−1isequivalenttothestatementthatallofthezerosoftheconfluenthypergeometricfunction0F1(c;z)arerealandnegativewhenc>0.However,itisknown[17]thatthezerosof0F1(c;z)arealsoreal(butnotnecessarilynegative)when−1Fromformulas(53)and(52)inBurchnallandChaundy[7]itfollowsthatifcisrealvaluedandc=0,−1,−2,...,thenwehavetheexpansionformulas(3.15)

|0F1(c;z)|=

2

1

(x2+y2)k0F1(c+2k;2x)

k=0k!(c)k(c)2k

7

∞󰀃

and(3.16)

(−1)j

x2j(0F1(c+2k+2j;x))2.0F1(c+2k;2x)=

j=0j!(c+2k+j−1)j(c+2k)2jAsintheBesselfunctioncase,substitute(3.16)into(3.15)andchangetheorderofsummationtoget

(−1)n

(3.17)|0F1(c;z)|=x2n

n=0n!(c+n−1)n(c)2n

2

∞󰀃∞󰀃

×2F1(−n,n+c−1;c;1+y2/x2)(0F1(c+2n;x))2

which,byapplyingthetransformationformula(3.7),gives(3.18)|0F1(c;z)|=

2

1

y2n

n=0n!(n+c−1)n(c)2n

×2F1(−n,1−n;c;1+x2/y2)(0F1(c+2n;x))2.

∞󰀃

Whenc>0andy=0thecoefficientof(0F1(c+2n;x))2intheseriesin

(3.18)isobviouslypositive.Hence,since0F1(c;x)>0whenc>0andx≥0,(3.18)givesanotherproofthat0F1(c;z)hasonlyrealnegativezeroswhenc>0.

Tohandlethecase−1∞󰀃∂(c+1)n22

(3.19)y|c0F1(c;z)|=2yy2n

∂yn=0n!(c+1)2n(c+1)2n+1

×2F1(−n,−n;c+1;1+x2/y2)(0F1(c+2n+2;x))2

and

(3.20)

∞󰀃∂2(c+1)n2

|cF(c;z)|=2y2n012∂yn=0n!(c+1)2n(c+1)2n+1

×2F1(−n,−n;c+1;1+x2/y2)(0F1(c+2n+2;x))2+4y

2

(c+2)n+1

y2n

n=0n!(c+1)2n+2(c+1)2n+3

∞󰀃

×2F1(−n,−n−1;c+2;1+x2/y2)(0F1(c+2n+4;x))2

8

which,inparticular,givetheinequalities

󰀆2∂󰀆󰀆

(3.21)y󰀆c(c+1)0F1(c;z)󰀆󰀆≥2(c+1)(y0F1(c+2;x))2,

∂y

c≥−1,

and(3.22)

∂222

|c(c+1)F(c;z)|≥2(c+1)(F(c+2;x)),0101∂y2c≥−1.

Sincethecoefficientsontherighthandsidesof(3.19)–(3.22)areclearlypositivewhenc>−1andy=0,theseformulasprovethatthefunctionsc(c+1)0F1(c;z)haveonlyrealzeroswhenc≥−1,whereitisunderstoodthatc(c+1)0F1(c;z)istobereplacedbyitsc→0limitcasez0F1(2;z)whenc=0,andbyitsc→−1limitcasez20F1(3;z)/2whenc=−1.

4

Laguerrepolynomialsand1F1(a;c;z)func-tions

Lαn(z)=

(α+1)n

1F1(−n;α+1;z)n!

Γ(n+α+1)

δnm,n,m=0,1,2,...,

n!

Whenα>−1theLaguerrepolynomials(4.1)

satisfytheorthogonalityrelation

󰀄

(4.2)

∞0

αα−x

Lαdx=n(x)Lm(x)xe

fromwhichitfollowsbyastandardargument(cf.[20,§3.3])thatthezeros

ofLαn(z)arerealandpositive.Analogoustothelastpartoftheprevioussection,inthissectionwewillderivesomesumsofsquaresexpansionswhich,besidesprovingtherealityofthezerosofthesepolynomialswhenα>−1,alsoprovethattheyhaveonlyrealzeros(notnecessarilypositive)when−1≥α≥−2,whereLαn(z)isdefinedtobetheα→−klimitcaseof(4.1)whenαisanegativeinteger−k.ThusLα1(z)=α+1−z,whichhasa󰀂negative

2

zerowhenα<−1,andLα2(z)=((α+1)(α+2)−2(α+2)z+z)2,whichhasnon-realzeroswhenα<−2.

9

Letαberealvalued.SubstitutingthesumofsquaresofLaguerrepoly-nomialsexpansion(from[7,(91)])(4.3)

+2kLαn−k(2x)

=

n−k󰀃j=0

(n−k−j)!(2k+2j+α)(2k+α)j

j!(2k+α)(2k+α+1)n+j−k

j2j

×(−1)x

󰀁

󰀅2

α+2k+2j

Ln−k−j(x)

intothespecialcaseof[3,(5.4)](4.4)

2

|Lαn(z)|

n

(α+1)n󰀃1+2k=(x2+y2)kLαn−k(2x)n!k=0k!(α+1)k

andchangingtheorderofsummationyields(4.5)

2

|Lα(z)|n

n

(α+1)n󰀃(n−k)!(2k+α)(α)k=(−1)kx2k

n!k!α(α+1)n+kk=0

×2F1(−k,k+α;α+1;1+y/x)

Thenapplicationof(3.7)gives(4.6)

2

|Lαn(z)|

n

(α+1)n󰀃(n−k)!(2k+α)(α)k2k=y

n!k!α(α+1)n+kk=0

22

󰀁

󰀅2

α+2k

Ln−k(x)

.

×2F1(−k,1−k;α+1;1+x/y)

22

󰀁

󰀅2

α+2k

Ln−k(x)

.

SinceLα0(x)≡1andthecoefficientsontherighthandsideof(4.6)areclearlypositivewhenα>−1andy=0,theexpansion(4.6)provesthattheLaguerrepolynomialshaveonlyrealzeroswhenα>−1.Thisalsofollows,inparticular,fromtheinequalities(4.7)

2

|Lαn(z)|≥

(α+1)n

y2n2F1(−n,1−n;α+1;1+x2/y2),

n!n!(n+α)n

α>−1,

and(4.8)

22α(z)|≥|L(x)|+|Lαnn

(α+1)n

y2n,

n!n!(n+α)n

10

α>−1,n≥1,

whichareconsequencesof(4.6).

Nowdifferentiateequation(4.5)withrespecttoyandapply(3.7)toderivetheexpansions

n−1󰀃∂(n−k−1)!(2k+α+2)(α+2)k2k2α2

(4.9)y|Ln(z)|=2yy

∂yn!k!(n+α+1)k+1k=0

×2F1(−k,−k;α+2;1+x/y)

and

22

󰀁

󰀅2

α+2k+2

Ln−k−1(x)

,n≥1,

(4.10)

n−1󰀃∂2(n−k−1)!(2k+α+2)(α+2)k2k2α

|L(z)|=2y∂y2nn!k!(n+α+1)k+1k=0

×2F1(−k,−k;α+2;1+x/y)+4y

2n−2󰀃

2

2

󰀁

󰀅2

α+2k+2

Ln−k−1(x)

(n−k−2)!(2k+α+4)(α+3)k+12k

y

n!k!(n+α+1)k+2k=0

2

2

×2F1(−k,−k−1;α+3;1+x/y)

whichyield,e.g.,theinequalities(4.11)and(4.12)

󰀁

󰀅2

α+2k+4

Ln−k−2(x),

n≥1,

∂2(α+2)󰀁α+2󰀅22α

y|Ln(z)|≥yLn−1(x),∂yn(n+α+1)2(α+2)󰀁α+2󰀅2∂22α

|L(z)|≥L(x),∂y2nn(n+α+1)n−1

α>−2,n≥1,

α>−2,n≥1,

andprove(afterlettingα→−2)thatthepolynomialsLαn(z)haveonlyreal

zeroswhenα≥−2.

Fortheconfluenthypergeometricfunctions1F1(a;c;z)withaandcrealvaluedandc=0,−1,−2,...,useoftheexpansionformulas[7,(42)and(43)]insteadof(4.3)and(4.4)yieldsthenonterminatingextensionof(4.5)(4.13)|1F1(a;c;z)|=

2

(a)k(c−a)k

x2k

k=0k!(c)2k(c+k−1)k

∞󰀃

×2F1(−k,c+k−1;c;1+y2/x2)(1F1(a+k;c+2k;x))2

11

andhence,by(3.7),(4.14)|1F1(a;c;z)|=

2

(a)k(c−a)k

(−1)ky2k

k=0k!(c)2k(c+k−1)k

∞󰀃

×2F1(−k,1−k;c;1+x2/y2)(1F1(a+k;c+2k;x))2.

Thendifferentiationofequation(4.13)withrespectofyandapplicationof

(3.7)givesthefollowingextensionsof(4.9)and(4.10)(andalsoof(3.19)and(3.20)),respectively,

(4.15)

∞󰀃∂(a)k+1(c−a)k+1(c+1)22

(−1)k+1y2ky|c(c+1)1F1(a;c;z)|=2y

∂yk=0k!(c+2)2k(c+k+1)k

×2F1(−k,−k;c+1;1+x2/y2)(1F1(a+k+1;c+2k+2;x))2

and

(4.16)

∞󰀃(a)k+1(c−a)k+1(c+1)∂22k+12k

|c(c+1)F(a;c;z)|=2(−1)y11∂y2k!(c+2)(c+k+1)2kkk=0

×2F1(−k,−k;c+1;1+x2/y2)(1F1(a+k+1;c+2k+2;x))2+4y

2

(a)k+2(c−a)k+2

(−1)ky2k

k=0k!(c+2)2k+2(c+k+3)k

∞󰀃

×2F1(−k,−k−1;c+2;1+x2/y2)(1F1(a+k+2;c+2k+4;x))2.

Ifa=−nisanegativeintegerandc=α+1,then(4.13)–(4.16)reduce

to(4.5),(4.6),(4.9),(4.10),respectively.Ifa=c+nwithnanonnegativeinteger,then(4.15)and(4.16)reducetoterminatingsumsofsquaresexpan-sionswithnonnegativecoefficientswhichprovethatc(c+1)1F1(c+n;c;z),asafunctionofz,hasonlyrealzeroswhenc≥−1,wherethisfunctionistobereplacedbyitsc→0andc→−1limitcaseswhenc=0andc=−1,respectively.Itshouldbenotedthat,inviewofKummer’stransformationformula[9,6.3(7)](4.17)

1F1(a;c;x)

=ex1F1(c−a;c;−x),

theseresultsonthezerosofc(c+1)1F1(c+n;c;z)areequivalenttothoseobtainedabovefortheLaguerrepolynomials.

12

5Jacobipolynomials

(α,β)

(z)=Pn

Whenα>−1andβ>−1theJacobipolynomials(5.1)

(α+1)n

2F1(−n,n+α+β+1;α+1;(1−z)/2)n!

satisfytheorthogonalityrelation

󰀄

(5.2)

1−1

(α,β)(α,β)Pn(x)Pm(x)(1−x)α(1+x)βdx=0,

n=m,

forn,m=0,1,2,...,andhence,by[20,Theorem3.3.1],thesepolynomials

haveonlyrealzeros.Inourderivationofsumsofsquaresexpansionswhichimplytherealityofthezerosofthesepolynomialswewillstartoutbyderivingsumsofsquaresexpansionsfornonterminating2F1(a,b;c;z)hypergeometricserieswith|z|<1(forconvergence).

Leta,b,cberealvalued,c=0,−1,−2,...,and|z|<1.Thenformula[6,(51)]givestheexpansion(5.3)

|2F1(a,b;c;z)|=

2

∞󰀃(a)k(b)k(c−a)k(c−b)kk=0

k!(c)k(c)2k

(x2+y2)k

×2F1(a+k,b+k;c+2k;2x−x2−y2).

Unfortunately,applicationoftheinversion[6,(50)]of[6,(51)]toeachofthe22

2F1(a+k,b+k;c+2k;2x−x−y)functionsontherightsideofequation(5.3)justreturnsonebacktothefunctionthatisontheleftside.Therefore,weuseformulas(44),(45),(50)in[6]toobtain,respectively,theexpansions(5.4)=

j=0

2F1(a

+k,b+k;c+2k;2x−x2−y2)

(−1)j(x2+y2)j2F1(a+k+j,b+k+j;c+2k+j;2x),

∞󰀃(a+k)j(b+k)j

j!(c+2k)j

(5.5)

2F1(a

+k+j,b+k+j;c+2k+j;2x)=

∞󰀃(a+k+j)m(b+k+j)mm=0

m!(c+2k+j)m

x2m

×2F1(a+k+j+m,b+k+j+m;c+2k+j+m;2x−x2),

13

(5.6)=

(a+k+j+m)n(b+k+j+m)n(c−a+k)n(c−b+k)n

(−1)nx2n

n!(c+2k+j+m+n−1)n(c+2k+j+m)2nn=0

2F1(a

∞󰀃

+k+j+m,b+k+j+m;c+2k+j+m;2x−x2)

×(2F1(a+k+j+m+n,b+k+j+m+n;c+2k+j+m+2n;x))2,andthensubstitutetheseexpansionsinturninto(5.3),changetheorderofsummationandusethebinomialtheoremtoobtain(5.7)

|2F1(a,b;c;z)|=

2

(a)m(b)m(c−a)j(c−b)j

(−1)mx2jy2m−2j

m=0j=0j!(m−j)!(c)m+j(m+c−1)j

m∞󰀃󰀃

×2F1(−j,m+c−1;c;1+y2/x2)(2F1(m+a,m+b;m+j+c;x))2.Applicationof(3.7)tothefirst2F1seriesontherightsideof(5.7)gives(5.8)|2F1(a,b;c;z)|=

2

(a)m(b)m(c−a)j(c−b)j

(−1)m+jy2m

m=0j=0j!(m−j)!(c)m+j(m+c−1)j

m∞󰀃󰀃

×2F1(−j,1−m;c;1+x2/y2)(2F1(m+a,m+b;m+j+c;x))2,

whichcontains(4.14)asalimitcase.Whena=−nisanegativeinteger,

b=n+α+β+1andc=α+1,itfollowsfrom(5.8)that(5.9)

=

󰀆󰀆2󰀆󰀆n!󰀆(α,β)

Pn(1−2z)󰀆󰀆󰀆

󰀆(α+1)n󰀆

mn󰀃󰀃(−n)m(n+α+β+1)m(n+α+1)j(−n−β)jm=0j=0

j!(m−j)!(α+1)m+j(m+α)j

(−1)m+jy2m

×2F1(−j,1−m;α+1;1+x2/y2)(2F1(m−n,m+n+α+β+1;m+j+α+1;x))2,

(α,β)

whichgivesasumsofsquaresproofthattheJacobipolynomialsPn(z)haveonlyrealzeroswhenα,β>−1(sincethecoefficientsin(5.9)arethenclearlypositive)andhence,bycontinuity,whenα,β≥−1.Therestriction

(α,β)

thatα,β≥−1cannotbeextendedtoα,β≥−2becauseP2(z)hasnon-realzeroswhenα,β>−2andα+β<−3.

Asinsections3and4onemayrepeatedlydifferentiate(5.7)withrespecttoyandapply(3.7)toobtainextensionsof(4.15),(4.16),etc.But,sincetheresultingidentitiesarequitelengthlyanddonotaddanyadditional(α,β)

14

forwhichtheJacobipolynomialshaveonlyrealzeros,wewillomitthemandonlypointoutthatthefirsttwodifferentiationsgiveidentitiesthatyield,inparticular,theinequalities(5.10)and(5.11)

󰀆2∂2󰀆2n(2n−1)(n+α+β+1)n(α+1)n2n−2󰀆(α,β)󰀆

yP(1−2z)≥󰀆󰀆

∂y2nn!n!

󰀆2∂󰀆2n(n+α+β+1)n(α+1)n2n󰀆(α,β)

y󰀆Pn(1−2z)󰀆y󰀆≥∂yn!n!

whenn≥1andα,β≥−1.

Insubsequentpapersitwillbeshownthatsquaresofrealvaluedfunctionscanalsobeusedtoprovetherealityofthezerosofsomenon-classicalfamiliesoforthogonalpolynomials,ofthecosinetransforms

󰀄

0∞

e−acoshtcosztdt,

a>0,

andofsomeotherentirefunctions.

References

[1]R.AskeyandG.Gasper,PositiveJacobipolynomialsumsII,Amer.J.Math.98

(1976),709–737.[2]R.AskeyandG.Gasper,Inequalitiesforpolynomials,inTheBieberbachConjecture,

ProceedingsoftheSymposiumontheOccasionoftheProof,SurveysandMonographs,No.21,Amer.Math.Soc.,Providence,RI(1986),7–32.[3]W.N.Bailey,OntheproductoftwoLegendrepolynomialswithdifferentarguments,

Proc.LondonMath.Soc.(2)41(1936),215–220.[4]R.P.Boas,EntireFunctions,AcademicPress,Inc.,NewYork,1954.

[5]L.deBranges,AproofoftheBieberbachconjecture,ActaMath.154(1985),137–152.[6]J.L.BurchnallandT.W.Chaundy,ExpansionsofAppell’sdoublehypergeometric

functions,Quart.J.Math.(Oxford)11(1940),249–270.[7]J.L.BurchnallandT.W.Chaundy,ExpansionsofAppell’sdoublehypergeometric

functions(II),Quart.J.Math.(Oxford)12(1941),112–128.

15

[8]L.Carlitz,Somepolynomialsrelatedtotheultrasphericalpolynomials,Portugaliae

Math.20(1961),127–136.[9]A.Erdelyi,HigherTranscendentalFunctions,vols.1and2,McGrawHill,NewYork,

1953.[10]G.Gasper,PositiveintegralsofBesselfunctions,SIAMJ.Math.Anal.6(1975),

868–881.[11]G.Gasper,Positivityandspecialfunctions,inTheoryandApplicationsofSpecial

Functions,R.Askey,ed.,AcademicPress,NewYork(1975),375–433.[12]G.Gasper,Positivesumsoftheclassicalorthogonalpolynomials,SIAMJ.Math.

Anal.8(1977),423–447.[13]G.Gasper,AshortproofofaninequalityusedbydeBrangesinhisproofofthe

Bieberbach,RobertsonandMilinconjectures,ComplexVariables:TheoryAppl.7(1986),45–50.[14]G.Gasper,q-ExtensionsofClausen’sformulaandoftheinequalitiesusedbyde

BrangesinhisproofoftheBieberbach,Robertson,andMilinconjectures,SIAMJ.Math.Anal.20(19),1019–1034.[15]G.Gasper,Usingsymboliccomputeralgebraicsystemstoderiveformulasinvolv-ingorthogonalpolynomialsandotherspecialfunctions,inOrthogonalPolynomials:TheoryandPractice,ed.byP.Nevai,KluwerAcademicPublishers,Boston,19,163–179.[16]G.GasperandM.Rahman,BasicHypergeometricSeries,CambridgeUniversity

Press,1990.[17]E.Hille,Noteonsomehypergeometricseriesofhigherorder,J.LondonMath.Soc.

4(1929),50–54.[18]J.L.W.V.Jensen,Recherchessurlath´eoriedes´equations,ActaMath.36(1913),

181–195.

¨[19]G.P´olya,Uberdiealgebraisch-funktionentheoretischenUntersuchungenvon

J.L.W.V.Jensen,Kgl.DanskeVidenskabernesSelskab.Math.-Fys.Medd.7(17)(1927),pp.3–33;reprintedinhisCollectedPapers,Vol.II,pp.278–308.[20]G.Szeg˝o,OrthogonalPolynomials,4thed.,Amer.Math.Soc.Colloq.Publ.23,Prov-idence,R.I.1975.[21]E.C.Titchmarsh,TheTheoryoftheRiemannZeta-Function,2ndedition(Revised

byD.R.Heath-Brown),OxfordUniv.Press,OxfordandNewYork,1986.[22]R.S.Varga,ScientificComputationonMathematicalProblemsandConjectures,

CBMS-NSFRegionalConferenceSeriesinAppliedMathematics,SIAM,Philadel-phia,1990.[23]G.N.Watson,TheoryofBesselFunctions,CambridgeUniv.Press,Cambridgeand

NewYork,1944.

16

GeorgeGasper

DepartmentofMathematicsNorthwesternUniversityEvanston,IL60208

E-Mail:g-gasper@nwu.edu

17